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href='/nlab/show/diff/natural+transformation'>natural transformation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a></p> </li> </ul> <h2 id='sidebar_universal_constructions'>Universal constructions</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal construction</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a>/<a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weighted+limit'>weighted limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/end'>end</a>/<a class='existingWikiWord' href='/nlab/show/diff/end'>coend</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kan+extension'>Kan extension</a></p> </li> </ul> </li> </ul> <h2 id='sidebar_theorems'>Theorems</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma'>Yoneda lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Isbell+duality'>Isbell duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+construction'>Grothendieck construction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor+theorem'>adjoint functor theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monadicity+theorem'>monadicity theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+lifting+theorem'>adjoint lifting theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tannaka+duality'>Tannaka duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Gabriel%E2%80%93Ulmer+duality'>Gabriel-Ulmer duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/small+object+argument'>small object argument</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freyd-Mitchell+embedding+theorem'>Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/relation+between+type+theory+and+category+theory'>relation between type theory and category theory</a></p> </li> </ul> <h2 id='sidebar_extensions'>Extensions</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sheaf+and+topos+theory'>sheaf and topos theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/enriched+category+theory'>enriched category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a></p> </li> </ul> <h2 id='sidebar_applications'>Applications</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/applications+of+%28higher%29+category+theory'>applications of (higher) category theory</a></li> </ul> <div> <p> <a href='/nlab/edit/category+theory+-+contents'>Edit this sidebar</a> </p> </div></div> <h4 id='limits_and_colimits'>Limits and colimits</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/limit'>limits and colimits</a></strong></p> <h2 id='1categorical'>1-Categorical</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/limit'>limit and colimit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/limits+and+colimits+by+example'>limits and colimits by example</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/commutativity+of+limits+and+colimits'>commutativity of limits and colimits</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/small+limit'>small limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/filtered+colimit'>filtered colimit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/directed+colimit'>directed colimit</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/sequential+colimit'>sequential colimit</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sifted+colimit'>sifted colimit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connected+limit'>connected limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/wide+pullback'>wide pullback</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/preserved+limit'>preserved limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/reflected+limit'>reflected limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/created+limit'>created limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a>, <a class='existingWikiWord' href='/nlab/show/diff/pullback'>fiber product</a>, <a class='existingWikiWord' href='/nlab/show/diff/base+change'>base change</a>, <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a>, <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a>, <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a>, <a class='existingWikiWord' href='/nlab/show/diff/cobase+change'>cobase change</a>, <a class='existingWikiWord' href='/nlab/show/diff/equalizer'>equalizer</a>, <a class='existingWikiWord' href='/nlab/show/diff/coequalizer'>coequalizer</a>, <a class='existingWikiWord' href='/nlab/show/diff/join'>join</a>, <a class='existingWikiWord' href='/nlab/show/diff/meet'>meet</a>, <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a>, <a class='existingWikiWord' href='/nlab/show/diff/initial+object'>initial object</a>, <a class='existingWikiWord' href='/nlab/show/diff/direct+product'>direct product</a>, <a class='existingWikiWord' href='/nlab/show/diff/direct+sum'>direct sum</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finite+limit'>finite limit</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/exact+functor'>exact functor</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kan+extension'>Kan extension</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+extension'>Yoneda extension</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weighted+limit'>weighted limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/end'>end and coend</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fibered+limit'>fibered limit</a></p> </li> </ul> <h2 id='2categorical'>2-Categorical</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-limit'>2-limit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inserter'>inserter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/isoinserter'>isoinserter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equifier'>equifier</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inverter'>inverter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/PIE-limit'>PIE-limit</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-pullback'>2-pullback</a>, <a class='existingWikiWord' href='/nlab/show/diff/comma+object'>comma object</a></p> </li> </ul> <h2 id='1categorical_2'>(∞,1)-Categorical</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-limit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-pullback'>(∞,1)-pullback</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>fiber sequence</a></li> </ul> </li> </ul> </li> </ul> <h3 id='modelcategorical'>Model-categorical</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+Kan+extension'>homotopy Kan extension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+product'>homotopy product</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+equalizer'>homotopy equalizer</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>homotopy fiber</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+pullback'>homotopy pullback</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+totalization'>homotopy totalization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coend'>homotopy end</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy colimit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coproduct'>homotopy coproduct</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coequalizer'>homotopy coequalizer</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cofiber+sequence'>homotopy cofiber</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+pushout'>homotopy pushout</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+realization'>homotopy realization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coend'>homotopy coend</a></p> </li> </ul> </li> </ul> <div> <p> <a href='/nlab/edit/infinity-limits+-+contents'>Edit this sidebar</a> </p> </div></div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#definition'>Definition</a></li><li><a href='#examples'>Examples</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#left_adjoints_to_constant_functors'>Left adjoints to constant functors</a></li><li><a href='#ConesOverTheIdentity'>Cones over the identity</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='definition'>Definition</h2> <p>\begin{definition} An <strong>initial object</strong> in a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a> <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>∅</mi></mrow><annotation encoding='application/x-tex'>\emptyset</annotation></semantics></math> such that for all objects <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mspace width='thinmathspace' /><mo>∈</mo><mspace width='thinmathspace' /><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>x \,\in\, \mathcal{C}</annotation></semantics></math>, there is a unique <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a> <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∅</mo><mover><mo>→</mo><mrow><mo>∃</mo><mo>!</mo></mrow></mover><mi>x</mi></mrow><annotation encoding='application/x-tex'>\varnothing \xrightarrow{\exists !} x</annotation></semantics></math> with <a class='existingWikiWord' href='/nlab/show/diff/source'>source</a> <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∅</mo></mrow><annotation encoding='application/x-tex'>\varnothing</annotation></semantics></math>. and <a class='existingWikiWord' href='/nlab/show/diff/target'>target</a> <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>. \end{definition}</p> <p>\begin{remark} An initial object, if it exists, is unique up to unique <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a>, so that we may speak of <a class='existingWikiWord' href='/nlab/show/diff/generalized+the'>the</a> initial object. \end{remark}</p> <p>\begin{remark}\label{InitialObjectIsEmptyColimit} When it exists, the initial object is the <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a> over the <a class='existingWikiWord' href='/nlab/show/diff/empty+diagram'>empty diagram</a>. \end{remark}</p> <p>\begin{remark} Initial objects are also called <em>coterminal</em>, and (rarely, though): <em>coterminators</em>, <em>universal initial</em>, <em>co-universal</em>, or simply <em>universal</em>. \end{remark}</p> <p>\begin{definition} An initial object <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∅</mo></mrow><annotation encoding='application/x-tex'>\varnothing</annotation></semantics></math> is called a <strong><a class='existingWikiWord' href='/nlab/show/diff/strict+initial+object'>strict initial object</a></strong> if all morphisms <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mover><mo>→</mo><mspace width='thickmathspace' /></mover><mo>∅</mo></mrow><annotation encoding='application/x-tex'>x \xrightarrow{\;} \varnothing</annotation></semantics></math> into it are <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphisms</a>.<br />\end{definition}</p> <p>\begin{remark} Initial objects are the <a class='existingWikiWord' href='/nlab/show/diff/duality'>dual</a> concept to <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal objects</a>: an initial object in <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is the same as a terminal object in the <a class='existingWikiWord' href='/nlab/show/diff/opposite+category'>opposite category</a> <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}</annotation></semantics></math>.<br />\end{remark}</p> <p>\begin{remark} An object that is both initial and <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal</a> is called a <a class='existingWikiWord' href='/nlab/show/diff/zero+object'>zero object</a>. \end{remark}</p> <h2 id='examples'>Examples</h2> <ul> <li> <p>An initial object in a <a class='existingWikiWord' href='/nlab/show/diff/partial+order'>poset</a> is a <a class='existingWikiWord' href='/nlab/show/diff/bottom'>bottom element</a>.</p> </li> <li> <p>The <a class='existingWikiWord' href='/nlab/show/diff/empty+set'>empty set</a> is an initial object in <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a>.</p> </li> <li> <p>Likewise, the <a class='existingWikiWord' href='/nlab/show/diff/empty+category'>empty category</a> is an initial object in <a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a>, the <a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty space</a> is an initial object in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a>, and so on.</p> </li> <li> <p>The <a class='existingWikiWord' href='/nlab/show/diff/trivial+group'>trivial group</a> is the initial object (in fact, the <a class='existingWikiWord' href='/nlab/show/diff/zero+object'>zero object</a>) of <a class='existingWikiWord' href='/nlab/show/diff/Grp'>Grp</a> and <a class='existingWikiWord' href='/nlab/show/diff/Ab'>Ab</a>.</p> </li> <li> <p>The <a class='existingWikiWord' href='/nlab/show/diff/ring'>ring</a> of <a class='existingWikiWord' href='/nlab/show/diff/integer'>integers</a> <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}</annotation></semantics></math> is the initial object of <a class='existingWikiWord' href='/nlab/show/diff/Ring'>Ring</a>.</p> </li> <li> <p>The <a class='existingWikiWord' href='/nlab/show/diff/field'>field</a> of <a class='existingWikiWord' href='/nlab/show/diff/rational+number'>rational numbers</a> <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Q}</annotation></semantics></math> is the initial object of <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Field</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>Field_0</annotation></semantics></math> (category of fields with <a class='existingWikiWord' href='/nlab/show/diff/characteristic'>characteristic</a> <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>) and the <a class='existingWikiWord' href='/nlab/show/diff/prime+field'>prime field</a> <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝔽</mi> <mi>p</mi></msub></mrow><annotation encoding='application/x-tex'>\mathbb{F}_p</annotation></semantics></math> is the initial object of <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Field</mi> <mi>p</mi></msub></mrow><annotation encoding='application/x-tex'>Field_p</annotation></semantics></math> (category of fields with characteristic <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math>), but none are the initial object of <a class='existingWikiWord' href='/nlab/show/diff/Field'>Field</a> (category of all fields), which actually doesn’t have one at all.</p> </li> <li> <p>The initial object of a <a class='existingWikiWord' href='/nlab/show/diff/under+category'>coslice category</a> <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo stretchy='false'>/</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>x/C</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/identity+morphism'>identity morphism</a> <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>→</mo><mi>x</mi></mrow><annotation encoding='application/x-tex'>x \to x</annotation></semantics></math>.</p> </li> <li> <p>An initial object in a category of <a class='existingWikiWord' href='/nlab/show/diff/central+extension'>central extensions</a> of a given algebraic object is called a <em><a class='existingWikiWord' href='/nlab/show/diff/universal+central+extension'>universal central extension</a></em>.</p> </li> </ul> <h2 id='properties'>Properties</h2> <h3 id='left_adjoints_to_constant_functors'>Left adjoints to constant functors</h3> <p>\begin{proposition} \label{AdjointsToConstantFunctors} Let <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a>.</p> <ol> <li> <p>The following are equivalent:</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> has a <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a>;</p> </li> <li> <p>the unique <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding='application/x-tex'>\mathcal{C} \to \ast</annotation></semantics></math> to the <a class='existingWikiWord' href='/nlab/show/diff/terminal+category'>terminal category</a> has a <a class='existingWikiWord' href='/nlab/show/diff/right+adjoint'>right adjoint</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow /></munder><mover><mo>⟵</mo><mrow /></mover></munderover><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'> \ast \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longleftarrow}} {\bot} \mathcal{C} </annotation></semantics></math></div></li> </ol> <p>Under this equivalence, the <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a> is identified with the image under the right adjoint of the unique object of the <a class='existingWikiWord' href='/nlab/show/diff/terminal+category'>terminal category</a>.</p> </li> <li> <p>Dually, the following are equivalent:</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> has an <a class='existingWikiWord' href='/nlab/show/diff/initial+object'>initial object</a>;</p> </li> <li> <p>the unique <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding='application/x-tex'>\mathcal{C} \to \ast</annotation></semantics></math> to the <a class='existingWikiWord' href='/nlab/show/diff/terminal+category'>terminal category</a> has a <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow /></munder><mover><mo>⟵</mo><mrow /></mover></munderover><mo>*</mo></mrow><annotation encoding='application/x-tex'> \mathcal{C} \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longleftarrow}} {\bot} \ast </annotation></semantics></math></div></li> </ol> <p>Under this equivalence, the <a class='existingWikiWord' href='/nlab/show/diff/initial+object'>initial object</a> is identified with the image under the left adjoint of the unique object of the <a class='existingWikiWord' href='/nlab/show/diff/terminal+category'>terminal category</a>.</p> </li> </ol> <p>\end{proposition}</p> <div class='proof'> <h6 id='proof'>Proof</h6> <p>Since the unique <a class='existingWikiWord' href='/nlab/show/diff/hom-set'>hom-set</a> in the <a class='existingWikiWord' href='/nlab/show/diff/terminal+category'>terminal category</a> is <a class='existingWikiWord' href='/nlab/show/diff/generalized+the'>the</a> <a class='existingWikiWord' href='/nlab/show/diff/singleton'>singleton</a>, the hom-isomorphism characterizing the <a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functors</a> is directly the <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a> of an <a class='existingWikiWord' href='/nlab/show/diff/initial+object'>initial object</a> in <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>(</mo><mo>*</mo><mo stretchy='false'>)</mo><mo>,</mo><mi>X</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><msub><mi>Hom</mi> <mo>*</mo></msub><mo stretchy='false'>(</mo><mo>*</mo><mo>,</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding='application/x-tex'> Hom_{\mathcal{C}}( L(\ast) , X ) \;\simeq\; Hom_{\ast}( \ast, R(X) ) = \ast </annotation></semantics></math></div> <p>or of a <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>R</mi><mo stretchy='false'>(</mo><mo>*</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><msub><mi>Hom</mi> <mo>*</mo></msub><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>,</mo><mo>*</mo><mo stretchy='false'>)</mo><mo>=</mo><mo>*</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> Hom_{\mathcal{C}}( X , R(\ast) ) \;\simeq\; Hom_{\ast}( L(X), \ast ) = \ast \,, </annotation></semantics></math></div> <p>respectively.</p> </div> <h3 id='ConesOverTheIdentity'>Cones over the identity</h3> <p>By definition, an initial object is equipped with a universal <a class='existingWikiWord' href='/nlab/show/diff/cocone'>cocone</a> under the unique functor <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>∅</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>\emptyset\to C</annotation></semantics></math> from the <a class='existingWikiWord' href='/nlab/show/diff/empty+category'>empty category</a>. On the other hand, if <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> is initial, the unique morphisms <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>!</mo><mo>:</mo><mi>I</mi><mo>→</mo><mi>x</mi></mrow><annotation encoding='application/x-tex'>!: I \to x</annotation></semantics></math> form a cone <em>over</em> the <a class='existingWikiWord' href='/nlab/show/diff/identity+functor'>identity functor</a>, i.e. a natural transformation <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Δ</mi><mi>I</mi><mo>→</mo><msub><mi>Id</mi> <mi>C</mi></msub></mrow><annotation encoding='application/x-tex'>\Delta I \to Id_C</annotation></semantics></math> from the <a class='existingWikiWord' href='/nlab/show/diff/constant+functor'>constant functor</a> at the initial object to the identity functor. In fact this is almost another characterization of an initial object (e.g. <a href='#MacLane'>MacLane, p. 229-230</a>):</p> <div class='num_lemma' id='cone'> <h6 id='lemma'>Lemma</h6> <p>Suppose <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>I\in C</annotation></semantics></math> is an object equipped with a natural transformation <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><mi>Δ</mi><mi>I</mi><mo>→</mo><msub><mi>Id</mi> <mi>C</mi></msub></mrow><annotation encoding='application/x-tex'>p:\Delta I \to Id_C</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mi>I</mi></msub><mo>=</mo><msub><mn>1</mn> <mi>I</mi></msub><mo>:</mo><mi>I</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>p_I = 1_I : I\to I</annotation></semantics></math>. Then <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> is an initial object of <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>.</p> </div> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>Obviously <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> has at least one morphism to every other object <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>X\in C</annotation></semantics></math>, namely <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'>p_X</annotation></semantics></math>, so it suffices to show that any <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>f:I\to X</annotation></semantics></math> must be equal to <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'>p_X</annotation></semantics></math>. But the naturality of <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> implies that <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='0em' rspace='thinmathspace'>Id</mo> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>∘</mo><msub><mi>p</mi> <mi>I</mi></msub><mo>=</mo><msub><mi>p</mi> <mi>X</mi></msub><mo>∘</mo><msub><mi>Δ</mi> <mi>I</mi></msub><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Id_C(f) \circ p_I = p_X \circ \Delta_I(f)</annotation></semantics></math>, and since <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mi>I</mi></msub><mo>=</mo><msub><mn>1</mn> <mi>I</mi></msub></mrow><annotation encoding='application/x-tex'>p_I = 1_I</annotation></semantics></math> this is to say <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>∘</mo><msub><mn>1</mn> <mi>I</mi></msub><mo>=</mo><msub><mi>p</mi> <mi>X</mi></msub><mo>∘</mo><msub><mn>1</mn> <mi>I</mi></msub></mrow><annotation encoding='application/x-tex'>f \circ 1_I = p_X \circ 1_I</annotation></semantics></math>, i.e. <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>=</mo><msub><mi>p</mi> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'>f=p_X</annotation></semantics></math> as desired.</p> </div> <div class='num_theorem' id='LimitOverIdentityFunctorIsInitialObject'> <h6 id='theorem'>Theorem</h6> <p>An object <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> in a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is initial iff <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a> of the <a class='existingWikiWord' href='/nlab/show/diff/identity+functor'>identity functor</a> <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Id</mi> <mi>C</mi></msub></mrow><annotation encoding='application/x-tex'>Id_C</annotation></semantics></math>.</p> </div> <div class='proof'> <h6 id='proof_3'>Proof</h6> <p>If <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> is initial, then there is a <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a> <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mo>!</mo> <mi>X</mi></msub><mo>:</mo><mi>I</mi><mo>→</mo><mi>X</mi><msub><mo stretchy='false'>)</mo> <mrow><mi>X</mi><mo>∈</mo><mi>Ob</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>(!_X: I \to X)_{X \in Ob(C)}</annotation></semantics></math> from <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Id</mi> <mi>C</mi></msub></mrow><annotation encoding='application/x-tex'>Id_C</annotation></semantics></math>. If <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>p</mi> <mi>X</mi></msub><mo>:</mo><mi>A</mi><mo>→</mo><mi>X</mi><msub><mo stretchy='false'>)</mo> <mrow><mi>X</mi><mo>∈</mo><mi>Ob</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>(p_X: A \to X)_{X \in Ob(C)}</annotation></semantics></math> is any cone from <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Id</mi> <mi>C</mi></msub></mrow><annotation encoding='application/x-tex'>Id_C</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mi>X</mi></msub><mo>=</mo><mi>f</mi><mo>∘</mo><msub><mi>p</mi> <mi>Y</mi></msub></mrow><annotation encoding='application/x-tex'>p_X = f \circ p_Y</annotation></semantics></math> for any <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>f:Y\to X</annotation></semantics></math>, and so in particular <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mi>X</mi></msub><mo>=</mo><msub><mo>!</mo> <mi>X</mi></msub><mo>∘</mo><msub><mi>p</mi> <mi>I</mi></msub></mrow><annotation encoding='application/x-tex'>p_X = !_X \circ p_I</annotation></semantics></math>. Since this is true for any <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mi>I</mi></msub><mo>:</mo><mi>A</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>p_I: A \to I</annotation></semantics></math> defines a morphism of cones, and it is the unique morphism of cones since if <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>q</mi></mrow><annotation encoding='application/x-tex'>q</annotation></semantics></math> is any morphism of cones, then <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mi>I</mi></msub><mo>=</mo><msub><mo>!</mo> <mi>I</mi></msub><mo>∘</mo><mi>q</mi><mo>=</mo><msub><mn>1</mn> <mi>I</mi></msub><mo>∘</mo><mi>q</mi><mo>=</mo><mi>q</mi></mrow><annotation encoding='application/x-tex'>p_I = !_I \circ q = 1_I \circ q = q</annotation></semantics></math> (using that <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>!</mo> <mi>I</mi></msub><mo>=</mo><msub><mn>1</mn> <mi>I</mi></msub></mrow><annotation encoding='application/x-tex'>!_I = 1_I</annotation></semantics></math> by initiality). Thus <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mo>!</mo> <mi>X</mi></msub><mo>:</mo><mi>I</mi><mo>→</mo><mi>X</mi><msub><mo stretchy='false'>)</mo> <mrow><mi>X</mi><mo>∈</mo><mi>Ob</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>(!_X: I \to X)_{X \in Ob(C)}</annotation></semantics></math> is the limit cone.</p> <p>Conversely, if <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>p</mi> <mi>X</mi></msub><mo>:</mo><mi>L</mi><mo>→</mo><mi>X</mi><msub><mo stretchy='false'>)</mo> <mrow><mi>X</mi><mo>∈</mo><mi>Ob</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>(p_X: L \to X)_{X \in Ob(C)}</annotation></semantics></math> is a limit cone for <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Id</mi> <mi>C</mi></msub></mrow><annotation encoding='application/x-tex'>Id_C</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>∘</mo><msub><mi>p</mi> <mi>Y</mi></msub><mo>=</mo><msub><mi>p</mi> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'>f\circ p_Y = p_X</annotation></semantics></math> for any <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>f:Y\to X</annotation></semantics></math>, and so in particular <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mi>X</mi></msub><mo>∘</mo><msub><mi>p</mi> <mi>L</mi></msub><mo>=</mo><msub><mi>p</mi> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'>p_X \circ p_L = p_X</annotation></semantics></math> for all <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. This means that both <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mi>L</mi></msub><mo>:</mo><mi>L</mi><mo>→</mo><mi>L</mi></mrow><annotation encoding='application/x-tex'>p_L: L \to L</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mn>1</mn> <mi>L</mi></msub><mo>:</mo><mi>L</mi><mo>→</mo><mi>L</mi></mrow><annotation encoding='application/x-tex'>1_L: L \to L</annotation></semantics></math> define morphisms of cones; since the limit cone is the terminal cone, we infer <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mi>L</mi></msub><mo>=</mo><msub><mn>1</mn> <mi>L</mi></msub></mrow><annotation encoding='application/x-tex'>p_L = 1_L</annotation></semantics></math>. Then by Lemma <a class='maruku-ref' href='#cone'>1</a> we conclude <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> is initial.</p> </div> <div class='num_remark' id='RelevanceForAdjointFunctorTheorem'> <h6 id='remark'>Remark</h6> <p><strong>(relevance for <a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor+theorem'>adjoint functor theorem</a>)</strong></p> <p>Theorem <a class='maruku-ref' href='#LimitOverIdentityFunctorIsInitialObject'>1</a> is actually a key of entry into the <a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor+theorem'>general adjoint functor theorem</a>. Showing that a functor <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>G: C \to D</annotation></semantics></math> has a <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> is tantamount to showing that each functor <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo stretchy='false'>(</mo><mi>d</mi><mo>,</mo><mi>G</mi><mo>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>D(d, G-)</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable</a>, i.e., that the <a class='existingWikiWord' href='/nlab/show/diff/comma+category'>comma category</a> <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo stretchy='false'>↓</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>d \downarrow G</annotation></semantics></math> has an initial object <math class='maruku-mathml' display='inline' id='mathml_0995af7e94ebc677459113c0d96923cf58729941_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>θ</mi><mo>:</mo><mi>d</mi><mo>→</mo><mi>G</mi><mi>c</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(c, \theta: d \to G c)</annotation></semantics></math> (see at <em><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functor</a></em>, <a href='#PointwiseExpressionOfLeftAdjoints'>this prop.</a>). This is the limit of the identity functor, but typically this is the limit over a large diagram whose existence is not guaranteed. The point of a solution set condition is to replace this with a small diagram which is cofinal in the large diagram.</p> </div> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a>, <a class='existingWikiWord' href='/nlab/show/diff/bi-terminal+object'>bi-terminal object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/empty+type'>bottom type</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/bi-initial+object'>initial object in a 2-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/terminal+object+in+a+quasi-category'>initial object in an (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/h-initial+object'>h-initial object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor+theorem'>adjoint functor theorem</a></p> </li> </ul> <h2 id='references'>References</h2> <p>Textbook accounts:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Francis+Borceux'>Francis Borceux</a>, Section 2.3 in Vol. 1: <em>Basic Category Theory</em> of: <em><a class='existingWikiWord' href='/nlab/show/diff/Handbook+of+Categorical+Algebra'>Handbook of Categorical Algebra</a></em>, Encyclopedia of Mathematics and its Applications <strong>50</strong> Cambridge University Press (1994) (<a href='https://doi.org/10.1017/CBO9780511525858'>doi:10.1017/CBO9780511525858</a>)</p> </li> <li id='MacLane'> <p><a class='existingWikiWord' href='/nlab/show/diff/Saunders+Mac+Lane'>Saunders MacLane</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Categories+for+the+Working+Mathematician'>Categories for the Working Mathematician</a></em></p> </li> </ul> <p> </p> </div> <div class="revisedby"> <p> Last revised on October 29, 2024 at 10:51:35. 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